At some points, we are going to make use of the correspondence between electric networks and random walks on weighted graphs (for reference see [1]}). A graph \(G=(V,E)\)  is said to be weighted if each edge \(e = (u,v) \in E\)  is assigned a weight \(w(e)\) . We say that the Markov chain \(X_0,X_1,\ldots \)  is a weighted random walk on \(G\)  if \({\mathbb {P}}(X_{t+1}=v|X_t=u) = w(u,v)/w(u)\) , where \(w(u)=\sum _{z:(u,z)\in E}w(u,z)\) .
There is valuable information to be found on this Markov chain by regarding \(G\)  as an electric network with each edge \(e\)  regarded as a resistor with conductance \(w(e)\)  and hence resistance \(1/w(e)\) . For \(u,v \in V\) , denote by \(R(u,v) = R_G(u,v)\)  the effective resistance between \(u\)  and \(v\)  in this electric network. Let \(m=m(G)= \sum _{e \in E}w(e)\)  be the total conductance of the graph.
For each vertex \(v\) , let \(T_v\)  be the first time that \(X\)  visits \(v\) , i.e. \(T_v=\min \lbrace t:X_t=v\rbrace \) . For vertices \(u,v\) , write
\(H(u,v)=H_G(u,v)={\mathbb {E}}_u[T_v]\)  for the hitting time of \(v\)  from \(u\) . The index \(u\)  to the expectation refers to conditioning on \(X_0=u\) .
The commute time between \(u\)  and \(v\)  is given by \(C(u,v)=C_G(u,v) = H_G(u,v)+H_G(v,u)\) .
Two well known facts follow.
